White light is used to illuminate the two slits in a Young's double slit experiment. The separation between the slits is b and the screen is at a distance D from the slits. At a point on a screen directly in front of one of the slits, certain wavelength are missing. Some of these wavelengths are:
(A) \( \lambda=\displaystyle\frac{b^{2}}{D} \)
(B) \( \lambda=\displaystyle\frac{2b^{2}}{D} \)
(C) \( \lambda=\displaystyle\frac{b^{2}}{3D} \)
(D) \( \lambda=\displaystyle\frac{2b^{2}}{3D} \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) and (C) only
A wavelength is missing at a point of the screen when the two waves arrive in opposite phase there, that is when the path difference is an odd multiple of \( \lambda/2 \).
For the point directly in front of one slit, the distances to the slits are D and \( \sqrt{D^{2} + b^{2}} \). Since \( D \gg b \), the binomial approximation gives:
\( \displaystyle \Delta = \sqrt{D^{2} + b^{2}} - D \approx \frac{b^{2}}{2D} \)
Setting this equal to \( (2m-1)\frac{\lambda}{2} \) for \( m = 1, 2, 3, \dots \):
\( \displaystyle \lambda = \frac{b^{2}}{(2m-1)D} = \frac{b^{2}}{D},\ \frac{b^{2}}{3D},\ \frac{b^{2}}{5D},\ \dots \)
(A) \( \lambda = b^{2}/D \) — Correct. This is the case \( m = 1 \).
(B) \( \lambda = 2b^{2}/D \) — Incorrect. It would need \( 2m - 1 = \frac{1}{2} \), which is not an odd whole number.
(C) \( \lambda = b^{2}/3D \) — Correct. This is the case \( m = 2 \).
(D) \( \lambda = 2b^{2}/3D \) — Incorrect. It would need \( 2m - 1 = \frac{3}{2} \), which is not an odd whole number.